Regular Sets Of Trees Research Articles

Free Kleene algebras with domain

First we identify the free algebras of the class of algebras of binary relations equipped with the composition and domain operations. Elements of the free algebras are pointed labelled finite rooted trees. Then we extend to the analogous case when the signature includes all the Kleene algebra with domain operations; that is, we add union and reflexive transitive closure to the signature. In this second case, elements of the free algebras are ‘regular’ sets of the trees of the first case. As a corollary, the axioms of domain semirings provide a finite quasiequational axiomatisation of the equational theory of algebras of binary relations for the intermediate signature of composition, union, and domain. Next we note that our regular sets of trees are not closed under complement, but prove that they are closed under intersection. Finally, we prove that under relational semantics the equational validities of Kleene algebras with domain form a decidable set.

The Cooper Storage Idiom

Cooper storage is a widespread technique for associating sentences with their meanings, used (sometimes implicitly) in diverse linguistic and computational linguistic traditions. This paper encodes the data structures and operations of cooper storage in the simply typed linear $$\lambda $$ -calculus, revealing the rich categorical structure of a graded applicative functor. In the case of finite cooper storage, which corresponds to ideas in current transformational approaches to syntax, the semantic interpretation function can be given as a linear homomorphism acting on a regular set of trees, and thus generation can be done in polynomial time.

Merging Hierarchically-Structured Documents in Workflow Systems

We consider the manipulation of hierarchically-structured documents within a complex workflow system. Such a system may consist of several subsystems distributed over a computer network. These subsystems can concurrently update partial views of the document. At some points in time we need to reconcile the various local updates by merging the partial views into a coherent global document. For that purpose, we represent the potentially-infinite set of documents compatible with a given partial view as a coinductive data structure. This set is a regular set of trees that can be obtained as the image of the partial view of the document by the canonical morphism (anamorphism) associated with a coalgebra (some kind of tree automaton). Merging partial views then amounts to computing the intersection of the corresponding regular sets of trees which can be obtained using a synchronization operation on coalgebras.

Flow analysis of lazy higher-order functional programs

In recent years much interest has been shown in a class of functional languages including HASKELL, lazy ML, SASL/KRC/MIRANDA, ALFL, ORWELL, and PONDER. It has been seen that their expressive power is great, programs are compact, and program manipulation and transformation is much easier than with imperative languages or more traditional applicative ones. Common characteristics: they are purely applicative, manipulate trees as data objects, use pattern matching both to determine control flow and to decompose compound data structures, and use a “lazy” evaluation strategy. In this paper we describe a technique for data flow analysis of programs in this class by safely approximating the behavior of a certain class of term rewriting systems. In particular we obtain “safe” descriptions of program inputs, outputs and intermediate results by regular sets of trees. Potential applications include optimization, strictness analysis and partial evaluation. The technique improves earlier work because of its applicability to programs with higher-order functions, and with either eager or lazy evaluation. The technique addresses the call-by-name aspect of laziness, but not memoization.

Une propri\\'et\\'e des forêts alg\\'ebriques « de greibach \\ra

Rounds has defined the class of context-free tree languages and proved its closure properties under linear homomorphism and intersection with regular sets of trees. The open problem of closure under linear inverse homomorphism has just been solved negatively. As for context-free grammars we can define Greibach's forms for context-free grammars of tree languages. Unfortunately there are context-free tree languages without Greibach's grammar. We define tree grammars without left (or on the top) recursivity and show they generate exactly the class of Greibach's tree languages. Then we prove the closure of that class under linear inverse homomorphism. This result proves that Greibach's tree languages are strictly contained in context-free tree languages.